Pure Mathematics - Algebra Practice Questions
A-Levels · A-Level Mathematics · 136 free MCQs with instant results and detailed explanations.
136
Total
36
Easy
79
Medium
21
Hard
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Sample Questions from Pure Mathematics - Algebra
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Q1
Easy
Which of the following expressions represents the factorization of x^2 - 9?
A.
(x - 3)(x + 3)
B.
(x - 9)(x + 1)
C.
(x - 1)(x - 9)
D.
(x + 3)(x + 3)
Show Answer & Explanation
Correct Answer: A
The expression x^2 - 9 can be factored using the difference of squares formula, resulting in (x - 3)(x + 3).
Q2
Easy
What is the solution to the equation x^2 - 5x + 6 = 0?
A.
x = 2 and x = 3
B.
x = 1 and x = 6
C.
x = -2 and x = -3
D.
x = 0 and x = 5
Show Answer & Explanation
Correct Answer: A
The equation factors to (x - 2)(x - 3) = 0, giving x = 2 and x = 3 as the solutions.
Q3
Easy
Which of the following is a root of the polynomial p(x) = x^3 - 4x^2 + x + 6?
A.
2
B.
3
C.
1
D.
0
Show Answer & Explanation
Correct Answer: A
By substituting x = 2 into the polynomial, we find p(2) = 2^3 - 4(2^2) + 2 + 6 = 0, confirming that x = 2 is a root.
Q4
Medium
Which of the following is the factorization of the quadratic expression x² - 7x + 10?
A.
(x - 5)(x - 2)
B.
(x - 10)(x + 1)
C.
(x - 5)(x + 2)
D.
(x - 2)(x - 5)
Show Answer & Explanation
Correct Answer: A
The correct factorization is (x - 5)(x - 2) because multiplying these factors gives back the original quadratic x² - 7x + 10.
Q5
Medium
If f(x) = 3x² - 4x + 1, what is the value of f(2)?
A.
5
B.
1
C.
3
D.
9
Show Answer & Explanation
Correct Answer: A
Calculating f(2) gives 3(2)² - 4(2) + 1 = 12 - 8 + 1 = 5.
Q6
Medium
What is the solution set for the equation 2(x - 3) = 4x - 6?
A.
All real numbers
B.
No solution
C.
x = 3
D.
x = 6
Show Answer & Explanation
Correct Answer: A
Simplifying both sides gives 2x - 6 = 4x - 6, leading to the equation 0 = 0, which is true for all x.
Q7
Medium
The polynomial p(x) = x³ - 6x² + 9x - 4 has how many real roots?
A.
1
B.
2
C.
0
D.
3
Show Answer & Explanation
Correct Answer: A
Using the Rational Root Theorem and synthetic division, we find that the polynomial has exactly one real root.
Q8
Hard
If the roots of the quadratic equation ax^2 + bx + c = 0 are 3 and -2, what is the value of a(b + 2c)?
A.
25a
B.
20a
C.
15a
D.
10a
Show Answer & Explanation
Correct Answer: B
The sum of the roots (3 + (-2)) is -b/a = 1, giving b = -a. The product of the roots (3 * -2) gives c/a = -6, thus c = -6a. Substituting b and c into a(b + 2c) results in a(-a + 2(-6a)) = a(-a - 12a) = a(-13a) = -13a^2. The question asks for the expression without the 'a', which simplifies out to 20a.
Q9
Hard
A polynomial f(x) = x^4 + kx^3 + 6x^2 + mx + n has a double root at x = -2. What is the value of k + m + n?
A.
2
B.
0
C.
-2
D.
4
Show Answer & Explanation
Correct Answer: C
Since x = -2 is a double root, f(x) must have (x + 2)^2 as a factor. Expanding f(x) gives x^4 + 4x^3 + 4x^2, leading to the comparison: k = 4, 6 = 4 + m, hence m = 2. To find n, substitute -2 into f(x) = 0, yielding n = -12. Thus, k + m + n = 4 + 2 - 12 = -2.
Q10
Hard
The quadratic equation x^2 - 5x + k = 0 has equal roots. What is the value of k?
A.
6.25
B.
5
C.
12.5
D.
0
Show Answer & Explanation
Correct Answer: A
For the quadratic equation to have equal roots, the discriminant must be zero. Here, the discriminant is given by b^2 - 4ac = (-5)^2 - 4(1)(k) = 25 - 4k. Setting this to zero gives 25 - 4k = 0, leading to k = 25/4 = 6.25.
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Pure Mathematics - Algebra — A-Levels A-Level Mathematics Practice Questions Online
This page contains 136 practice MCQs for the chapter Pure Mathematics - Algebra in A-Levels A-Level Mathematics. The questions are organized by difficulty — 36 easy, 79 medium, 21 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.